Spring model

We refer to this model as the bead-spring model and to its theoretical development as the Rouse theory, although Rouse, Bueche, and Zimm have all been associated with its development. 

The bead and spring model is clearly based on mechanical elements just as the Maxwell and Voigt models were. There is a difference, however. The latter merely describe a mechanical system which behaves the same as a polymer sample, while the former relates these elements to actual polymer chains. As a mechanical system, the differential equations represented by Eq. (3.89) have been thoroughly investigated. The results are somewhat complicated, so we shall not go into the method of solution, except for the following observations ... 

Figure 1.11(b) illustrates the ball-and-spring model which is adequate for an approximate treatment of the vibration of a diatomic molecule. For small displacements the stretching and compression of the bond, represented by the spring, obeys Hooke s law ... 

The relationship between fiber and matrix moduH and fiber volume fraction for a unidirectional lamina loaded in the direction transverse to the fibers is not simple. A lower bound (1) is given by the expression of the series spring model. 

An estimate of the shear modulus is also given by an expression based on the series spring model... 

Demonstrations (a) Atom spring models (Fig. 4.2) on overhead projector to illustrate effect of structure on modulus, (b) Large models of Na atom and Cl atom, (c) Liquid nitrogen. 

Demonstration Fit up a dashpot and spring model (Fig. 19.8) and hang it from a support. Hang a weight on the lower end of the combination and, using a ruler to measure extension, plot the creep out on the blackboard. Remove weight and plot out the reverse creep. 

Since this behavior is universal, it is obvious that the simplest simulation models which contain the essential aspects of polymers are sufficient to study these phenomena. Two typical examples of such models are the bond fluctuation Monte Carlo model and the simple bead-spring model employed in molecular dynamics simulations. Both models are illustrated in Fig. 6. 

Concluding this section, one should mention also the method of molecular dynamics (MD) in which one employs again a bead-spring model [33,70,71] of a polymer chain where each monomer is coupled to a heat bath. Monomers which are connected along the backbone of a chain interact via Eq. (8) whereas non-bonded monomers are assumed usually to exert Lennard-Jones forces on each other. Then the time evolution of the system is obtained by integrating numerically the equation of motion for each monomer i... 

Again, the OLMC bead-spring model (Sec. IIB 2) is used, with a host matrix of an equilibrated dense solution of polymer chains quenched at different concentrations Cots. Eq. (7) for the probability IF of a random monomer displacement in direction Ax, Ay, Az is given by... 

A. Milchev, K. Binder. Static and dynamic properties of adsorbed chains at surfaces Monte Carlo simulations of a bead-spring model. Macromolecules 29 343-354, 1996. 

Bead-spring models without explicit solvent have also been used to simulate bilayers [40,145,146] and Langmuir monolayers [148-152]. The amphi-philes are then forced into sheets by tethering the head groups to two-dimensional surfaces, either via a harmonic potential or via a rigid constraint. 

The inclusion of internal viscosity raises considerably the free-energy storage capacity of a rapidly deforming macromolecule as compared to the idealized Hookean spring model and could play a decisive role in mechanochemical reactivity in transient elongational flow. 

Analysis of the prototypical resonant swing spring model [11-13] shows that Fermi resonance with conserved angular momentum is an intrinsically three-dimensional phenomenon. The form of the 3x3 monodromy matrix was given. 

First approaches at modeling the viscoelasticity of polymer solutions on the basis of a molecular theory can be traced back to Rouse [33], who derived the so-called bead-spring model for flexible coiled polymers. It is assumed that the macromolecules can be treated as threads consisting of N beads freely jointed by (N-l) springs. Furthermore, it is considered that the solution is ideally dilute, so that intermolecular interactions can be neglected. 

Zimm [34] extended the bead-spring model by additionally taking hydrodynamic interactions into account. These interactions lead to changes in the medium velocity in the surroundings of each bead, by beads of the same chain. It is worth noting that neither the Rouse nor the Zimm model predicts a shear rate dependency of rj. Moreover, it is assumed that the beads are jointed by an ideally Hookean spring, i.e. they obey a strictly linear force law. 

The molecular mechanics method, often likened to a ball and spring model of the molecule, represents the total energy of a system of molecules with a set of simple analytical functions representing different interactions between bonded and non-bonded atoms, as shown schematically in Figure 1. 

In a study by da Silva et al. (1988), the hydrogen was assumed to be in the X—AB position. They constructed a spring model of this structure and fit the spring constants to demonstrate that experimentally measured frequencies could be produced for H—B, H—Al, and H—Ga pairs in the X—AB configuration. Although their original study described the electronic structure in terms of SW-Xa-cluster calculations, these vibrational fits were produced from a classical model. 

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